RE: connectedness

["Marta Bunge" <martabunge@hotmail.com>]

Dear Paul,

>First, though, I would like to underline something that Steve Lack
>(almost) said, namely that the category in which you index your
>components, and therefore also the one in which you define
>connectedness, need to be EXTENSIVE, ie their coproducts should
>be disjoint, and stable under pullback, and the initial object strict.
>
>Maybe we've over-done philology recently, but "component" means
>"putting together", where we expect the parts to cover the whole
>(coproduct), without overlapping (disjoint), to be distinguishable
>(like disjoint union, but unlike addition and disjunction).
>The modern notion of extensivity, in which Steve had a part,
>captures this idea very neatly.   The equivalence between definitions
>of connectedness based on 1+1 and on X+Y surely depends on stability
>under pullback, and the requirement that the choice between left
>and right be unique surely requires disjointness.   Maybe a close
>study of Marta Bunge's work on abstract connectedness would clarify
>this.
>

In my now obsolete 1966 thesis, the context was that of a category with
finite limits and finite coproducts, but I had not assumed therein that
coproducts should be disjoint and universal. With these assumptions, the
definitions of `abstractly unary' for arbitrary binary products (factors
through AT LEAST one of the injections) and of `abstractly exclusively
unary' (factors through exactly one of the injections) are equivalent by
the disjointness part, and are equivalent also to the same notions with
binary coproducts of 1 instead of arbitrary binary coproducts (by the
universal or stability part). So, *any* of those in this context should
mean `connected'.  Without those conditions, but with just a terminal
object and binary coproducts, then the `at least' part does not reduce to
that of coproducts of 1, but the `at most' part does. In that case, the
notions correspond to `abstractly unary' and `abstractly exclusively
unary', and they are not equivalent. So it all depends on the ambient
category.

>
>So far, I have only mentioned BINARY notions of connectedness,
>but if we want to talk about families of connected COMPONENTS
>then we must also consider INFINITARY connectedness (as Marta
>stressed).  Here the results for the constructive real line are
>somewhat surprising.
>
>
I still have to digest your disquisitions on constructive analysis, which
seem most interesting, but on the above point, I emphasize that inded one
must keep the disctinction between connectedness wirt binary coproducts and
connectedness wrt arbitrary coproducts (indexed externally, e.g. by a set in
a Grothendieck topos E-->SET, or by an object of S in the cae of a bounded
topos E-->S. Whether the terminology must make that distinction I am not
sure of, or maybe we could say `connected' for the binary case (which is
also intrinsic), and `S-connected' for the case of S-indexed coproducts.
Once again, under enough hypotheses as I szaid above and was also mentioned
by Steve Lack.

I am not sure of which hypotheses Vaughan wants to make but, if the minimal
possible, then he might need the detailed analysis that I proposed and, in
that case, reserve `connected' in the binary case for `abstractly
exclusively unary', not simply `abstractly unary', and `connected' when only
the binary coproduct considered is 1+1. But if his categories ae categories
of graphs, I don't see his problem.


With best regards,
Marta